To simplify the given expression:

$\frac{3^{2n+3} - 18(3)^{2(n-1)}}{5(3^n)^2}$

we can break down each term and simplify step-by-step.

Step 1: Rewrite Terms with Exponents

The expression in the numerator is: $3^{2n+3} - 18 \cdot 3^{2(n-1)}$ and the denominator is: $5(3^n)^2$

Let's rewrite $(3^n)^2$ in the denominator as $3^{2n}$, giving us: $\frac{3^{2n+3} - 18 \cdot 3^{2n-2}}{5 \cdot 3^{2n}}$

Step 2: Factor Out Common Terms in the Numerator

In the numerator, both terms have a factor of $3^{2n}$: $3^{2n+3} = 3^{2n} \cdot 3^3$ $18 \cdot 3^{2n-2} = (18 \cdot 3^{-2}) \cdot 3^{2n} = 2 \cdot 3^{2n}$

So, rewrite the numerator as: $3^{2n} \cdot (3^3 - 2)$

This gives us: $\frac{3^{2n} \cdot (27 - 2)}{5 \cdot 3^{2n}} = \frac{3^{2n} \cdot 25}{5 \cdot 3^{2n}}$

Step 3: Simplify by Canceling $3^{2n}$

Now, $3^{2n}$ cancels out in both the numerator and denominator, leaving: $\frac{25}{5} = 5$

Final Answer:

$5$

Would you like further details or have any questions about this solution?

Here are five additional questions related to simplifying expressions with exponents and factoring:

1. How would you simplify an expression with a negative exponent in both the numerator and denominator?
2. What are the general rules for simplifying expressions with multiple exponent terms?
3. How can you use factoring to simplify expressions with large coefficients and exponents?
4. How would this problem change if the exponent bases were different (e.g., 2 and 3 instead of just 3)?
5. How do you approach simplification when terms have mixed operations, like addition and multiplication?

Tip: Always look for common factors or terms when simplifying complex expressions; it can make the process much more manageable.